The content of my posting of 5:28PM 9/11/98 "f.o.m. and whores" seems to
have resonated well with a number of people. I chose the most attention
grabbing terminology that came to mind at the time. But now I want to
upgrade that terminology.
This is a repeat of that posting incorporating this change of terminology.
I suspect that the terminology can be improved further.
*****
Hersh 1:22PM 9/11/98 writes:
> In describing foundationalism in the sense of Lakatos,
>(Frege, Russell, Hilbert, Brouwer) I did not consider or have
>in mind the research in logic and set theory of Harvey Friedman
>and his circle. I do not regard this as part of foundationalism,
>because the restoration of indubitability is not its goal.
>It seems to me to be part of logic and set theory, which are part of
>mathematics. For whatever reasons, Friedman says it is not
>part of mathematics. In any case, whatever I wrote about
>foundationalism has nothing to do with Friedmans "foundations of
>mathematics."
I appreciate this comment, but would like to amplify on it and clarify some
points.
My conception of "foundations of mathematics" refers to a line of research,
or a style of intellectual inquiry, and not to any point of view on any
philosophical issues. On the contrary, it is agnostic on philosophical
issues in the following extreme sense: any even remotely sensible
philosophical view, however weak or strong, is material for investigation.
The goal is to make sufficient sense out of whatever it is in order to
create new science of hopefully "permanent value" (see below) which can be
interpreted to attack or defend that philosophical view. For instance, we
all know that Godel's incompleteness theorems can and have been used to
attack and defend all sorts of competing and mutually incompatible
philosophical views. (Here I mean science in the **broadest possible
sense** of systematic subjects). This represents a kind of "philosophical
agnosticism."
>From time to time, some people have set me aside and said: forget about new
science, tell me what you REALLY think. I invariably respond: but I'm not
ready.
Closer to the truth is that I have become addicted to the exquisite
pleasures of doing this kind of thing for 30 years. One hopefully develops
a nose for sensing the kind of scientific investigations that are, or can,
or will become essential to an informed discussion of a wide variety of
philosophical issues. This is the "permanent value" referred to above.
I don't want to really say anything like "foundations of mathematics is not
mathematics." What I want to say is that "foundations of mathematics is not
ONLY mathematics," and in my more expansive moods, I want to say that
"foundations of mathematics is not PRIMARILY mathematics."
Of course, I don't really know what mathematics *really is* in any deep
sense; I have not even spent the requisite time and effort to try to figure
this out. So in a sense, I have no idea what the previous paragraph
*really* means.
So instead I want to emphasize the **operational** aspect of that
paragraph. Namely, "foundations of mathematics should not be judged solely
by mathematicians". Or, in more expansive moods, "foundations of
mathematics should not be primarily judged by mathematicians". And in my
most expansive moods, "foundations of mathematics should be primarily
judged by nonmathematicians." (This paragraph is in no way related to
mathematician Hersh or any writings or views of Hersh about foundations
of mathematics in the sense I have used here or any other. I am referring
to the general mathematics community).